Given that $\cos \theta =\frac{5}{13}$ and the angle θ lies in the fourth quadrant, determine the value of $\sin \theta$.

The correct answer is:

$-\frac{12}{13}$

Step 1: Use the Pythagorean trigonometric identity

The fundamental trigonometric identity relating sine and cosine is:

${\sin }^2\theta +{\cos }^2\theta =1$

Rearranging the equation to solve for
${\sin }^2\theta$
:

${\sin }^2\theta =1-{\cos }^2\theta$

Step 2: Substitute the given cosine value

We are given that:

$\cos \theta =\frac{5}{13}$

Substitute this value into the identity:

${\sin }^2\theta =1-{\left(\frac{5}{13}\right)}^2$

${\sin }^2\theta =1-\frac{25}{169}$

${\sin }^2\theta =\frac{169-25}{169}$

${\sin }^2\theta =\frac{144}{169}$

Step 3: Take the square root and select the correct sign

Taking the square root of both sides gives:

$\sin \theta =\pm \sqrt{\frac{144}{169}}=\pm \frac{12}{13}$

Since the angle
$\theta$
lies in the fourth quadrant, the sine function must be negative (
$\sin \theta <0$
).

Therefore, selecting the negative value gives:

$\sin \theta =-\frac{12}{13}$